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Try to solve this problem number 35 from Singapore Mathematics Olympiad, SMO, 2018 based on Functional Equation.

Consider integers \({1,2, \ldots, 10}\). A particle is initially -at 1 . It moves to an adjacent integer in the next step. What is the expected number of steps it will take to reach 10 for the first time?

- 82
- 81
- 80
- 79

Functional Equation

Equation

But try the problem first...

Answer : 81

Source

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Singapore Mathematical Olympiad

Challenges an Thrills - Pre - College Mathematics

First hint

If you got stuck into this problem we can start taking an expected number of steps to be \(g_{n}\). We need to remember at first the particle was in 1 then it will shift to the next step so for n no of position we can expressed it as n and n -1 where n = 2,3,4,........,100.

Now try the rest..............

Second Hint

Now let's continue after the last hint ............

Then \(g_{n+1} = \frac {1}{2} (1+g_{n} + g_{n+1} )+ \frac {1}{2}\)

which implies , \(g_{n+1} = g_{n} + 2\)

Now we know that,\(g_{2} = 1\). Then \(g_{3} = 3\), \(g_{4}= 5\),..................,\(g_{10}=17\)

\(g = g_{2}+g_{3}+g_{4}+....................+g_{10} = 1+3+.....................+17 = 81\)[ Answer]

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